# Exploring Treatment Strategies for Type 1 Diabetes through Mathematical Modelling: Updating and Expanding an Existing Model

Kenneth Hagde Mandrup Nielsen

Studenteropgave: Speciale

## Abstrakt

In this thesis we start out by dealing with a mathematical model of the onset of type 1 diabetes as proposed by Marée et al. (2006). It is a 5-dimensional model that uses ordinary differential equations (ODEs) to simulate the behavior of macrophages, activated macrophages, apoptotic beta-cells, necrotic beta-cells, and cytotoxic cytokines in neonate NOD- and Balb/c-mice. The purpose of Marée et al. (2006) is primarily to investigate the hypothesis that impaired macrophage phagocytosis can mean the difference between health and type 1 diabetes (in mice). Marée et al. (2006) base their hypothesis on previous work by Trudeau et al. (2003) and Marée et al. (2005). We start out by presenting an introduction to the biology behind the DuCa model and introduce some of the tentative treatment strategies as of today. With the groundwork in place we present, and discuss the foundations of, the model, which we have dubbed the DuCa model, and provide an analysis of a reduced version of the DuCa model, called the intermediate model (IM). The IM is based on the so-called Copenhagen model, made by Blasio et al. (1999). After this gentle interlude we proceed to do a codimension 1 bifurcation analysis of the DuCa model. The analysis serves to determine the overall soundness of the model. Where by “soundness” we mean the lack of nonphysiological behavior within a reasonable range of key parameters. The bifurcation analysis, and a thorough discussion of the adherent assumptions as well as simplifications, leads us to conclude that the DuCa model is sound. Based on these findings we expand the DuCa model, guided by recent data and guidelines that should apply to all mathematical models as well as some criteria that pertain to this model in particular. The expansion consists of a compartment of healthy beta-cells. By this expansion we add to the realism of the model and set up a model from which future researches can analyze how best to go about countering the chronic inflammation of the Islets of Langerhans that leads to T1D in 80 % of female NOD-mice. In our discussion of the expanded model we also provide some hints as to how one would go about implementing the effect of, in particular, one tentative drug, as well as point to additional features that need implementing (and how this could be done).

Uddannelser Matematik, (Bachelor/kandidatuddannelse) Kandidat Engelsk 6 mar. 2010 Johnny T. Ottesen & Flemming Pociot

### Emneord

• Type 1 Diabetes
• ODE
• Treatment
• compartment model
• IDDM
• Bifurcation analysis
• codimension one analysis
• System biology
• Mathematical modelling